Exponential lower bound for the eigenvalues of the time-frequency localization operator before the plunge region

For a pair of sets $T,\Omega \subset R$ the time-frequency localization operator is defined as $S_{T,\Omega }=P_T{F}^{-1}{P}_{\Omega }F{P}_T$, where $F$ is the Fourier transform and $P_T,P_{\Omega }$ are projection operators onto T and Ω, respectively. We show that in the case when both T and Ω are intervals, the eigenvalues of $S_{T,\Omega }$ satisfy ${\lambda }_n(T,\Omega )\ge 1-{\delta }^{\left|T\right|\left|\Omega \right|}$ if $n\le (1-\epsilon )\left|T\right|\left|\Omega \right|$, where $\epsilon >0$ is arbitrary and $\delta =\delta \left(\epsilon \right)<1$, provided that $\left|T\right|\left|\Omega \right|>c_{\epsilon }$. This improves the result of Bonami, Jaming and Karoui, who proved it for $\epsilon \ge 0.42$. The proof is based on the properties of the Bargmann transform.